I have some interest for the following inequality :
$$\pi ^{-e}-e^{-\pi}> \frac{e^{\frac{1}{e}-e-\pi}}{\pi}$$
I have tried to study the following function : $$f(x)=x^{-y}-y^{-x}$$ And after use convexity but it's really too sharp to this kind of reasoning .
So I have used power series to get big polynomials .Unfortunatly I can't solve this way.
Furthermore the inequality :
Let $x>y>0$ then we have : $$x ^{-y}-y^{-x}> \frac{y^{\frac{1}{y}-y-x}}{x}$$
is false.
My question :How to solve it ?
So if you have great ideas...
...Thanks a lot.